[petsc-dev] Runge-Kutta example in TS

[Jed Brown]

On Thu, Jan 17, 2013 at 5:49 PM, Gautam Bisht <gbisht at lbl.gov> wrote:

> Thanks guys for your pointers.
>
> I was wondering if for explicit scheme, TS can increase/decrease 'dt' and
> point out if stability condition is violated or not.
>

There is no practical automatic way to determine that the "stability
condition" has been violated. A combination of the linear stability region
and eigenanalysis of the operator would be sufficient in many cases, but
that is very expensive to compute and still isn't right in the presence of
strong nonlinearity. For hyperbolic problems, you can evaluate the CFL
criteria to determine a necessary (but not necessarily sufficient)
condition for stability.

The accuracy-based controllers can sometimes be used, but they result in
many step rejections for "stiff" problems (loosely meaning any problem
where step size is limited by stability rather than accuracy; thus
dependent on error tolerances here). The reason is that the solution is
very smooth when close to the slow manifold, so the controller takes larger
steps until an (exponentially growing) instability exceeds the tolerance,
causing the step to shorten.
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